   Chapter 9.5, Problem 29ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# An alternative way to derive Theorem 9.5.1 uses the following division rule: Let n and k be integers so that k divides n. If a set consisting of n elements is divided into subsets that each contain k elements, then the number of such subsets is n/k. Explain how Theorem 9.5.1 can be derived using the division rule.

To determine

To explain:

The theorem (nr)=n!r!(nr)! using the division rule.

Explanation

Given information:

Division Rule: Let n and k be integers so that k divides n. If a set consisting of n elements is divided into subsets that each contains k elements, then the number of such subsets is nk.

Concept used:

Now, consider a set of n elements, out of which a subset of size r is required to calculate.

Now, consider mapping any permutation of an n -element set (a1,a2,a3,....,an) into k -element set.

Such mapping is done by taking the first k elements of the permutation.

Notice that any other permutation with the same first k elements a1,a2,a3,....,ak in any order and the same remaining elements nk in any order will also map to this set.

Calculation:

Now there are r! possible permutations of the first r elements.

Also (nr)! permutations are possible for the remaining nr elements

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